On simultaneous representations of primes by binary quadratic forms

Joseph B. Muskat

doi:10.1016/0022-314X(84)90111-2

On simultaneous representations of primes by binary quadratic forms

Joseph B. Muskat

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M² + 7N², and knowing M and N makes it possible to "predict" whether p = A² + 14B² is solvable or p = 7C² + 2D² is solvable. More generally, let q and r be distinct primes, and let an integral solution of H²p = M² + qN² be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K²q = A² + qrB² is solvable and the possible values of K′ for which K′²p = qC² + rD² is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant -4qr from representing p.

Original language	English
Pages (from-to)	263-282
Number of pages	20
Journal	Journal of Number Theory
Volume	19
Issue number	2
DOIs	https://doi.org/10.1016/0022-314X(84)90111-2
State	Published - Oct 1984

Bibliographical note

Funding Information:
* This research received some support from National

Funding

* This research received some support from National

Funders	Funder number
Russian Science Foundation	GP8973

Access to Document

10.1016/0022-314X(84)90111-2

Cite this

@article{280a0c4381d448958f56a8dbfd35dacf,

title = "On simultaneous representations of primes by binary quadratic forms",

abstract = "Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M2 + 7N2, and knowing M and N makes it possible to {"}predict{"} whether p = A2 + 14B2 is solvable or p = 7C2 + 2D2 is solvable. More generally, let q and r be distinct primes, and let an integral solution of H2p = M2 + qN2 be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K2q = A2 + qrB2 is solvable and the possible values of K′ for which K′2p = qC2 + rD2 is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant -4qr from representing p.",

author = "Muskat, {Joseph B.}",

note = "Funding Information: * This research received some support from National",

year = "1984",

month = oct,

doi = "10.1016/0022-314X(84)90111-2",

language = "אנגלית",

volume = "19",

pages = "263--282",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press Inc.",

number = "2",

}

TY - JOUR

T1 - On simultaneous representations of primes by binary quadratic forms

AU - Muskat, Joseph B.

N1 - Funding Information: * This research received some support from National

PY - 1984/10

Y1 - 1984/10

N2 - Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M2 + 7N2, and knowing M and N makes it possible to "predict" whether p = A2 + 14B2 is solvable or p = 7C2 + 2D2 is solvable. More generally, let q and r be distinct primes, and let an integral solution of H2p = M2 + qN2 be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K2q = A2 + qrB2 is solvable and the possible values of K′ for which K′2p = qC2 + rD2 is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant -4qr from representing p.

AB - Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M2 + 7N2, and knowing M and N makes it possible to "predict" whether p = A2 + 14B2 is solvable or p = 7C2 + 2D2 is solvable. More generally, let q and r be distinct primes, and let an integral solution of H2p = M2 + qN2 be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K2q = A2 + qrB2 is solvable and the possible values of K′ for which K′2p = qC2 + rD2 is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant -4qr from representing p.

UR - http://www.scopus.com/inward/record.url?scp=37049021388&partnerID=8YFLogxK

U2 - 10.1016/0022-314X(84)90111-2

DO - 10.1016/0022-314X(84)90111-2

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:37049021388

SN - 0022-314X

VL - 19

SP - 263

EP - 282

JO - Journal of Number Theory

JF - Journal of Number Theory

IS - 2

ER -

On simultaneous representations of primes by binary quadratic forms

Abstract

Bibliographical note

Funding

Access to Document

Other files and links

Fingerprint

Cite this